A proper push-up takes you all the way down to the ground -- but it's easy to cheat by only going partway down. To be sure you're getting the full challenge and benefit out of each rep, pick up your hands every time your chest hits the floor.
At the beginning of January, I happened upon a video you may have seen: A young woman doing push-ups for 100 days. She starts out "scrawny", but gets stronger. I wanted to do that. So I did, and you can too.